Autonomous multiple teams establishment for mobile sensor networks by SVMs within a potential field

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Autonomous multiple teams establishment for mobile sensor networks

by SVMs within a potential ﬁeld

Sedat Nazlibilek

⇑

Atilim University, Engineering Faculty, Department of Mechatronics, Ankara, Turkey Bilkent University, NANO Technology Research Center, 06800 Ankara, Turkey

a r t i c l e

i n f o

Article history: Received 23 June 2011

Received in revised form 13 October 2011 Accepted 31 January 2012

Available online 15 February 2012 Keywords:

Algorithms Measurement Networks

Magnetic ﬁeld measurement Robots

Vectors

a b s t r a c t

In this work, a new method and algorithm for autonomous teams establishment with mobile sensor network units by SVMs based on task allocations within a potential ﬁeld is proposed. The sensor network deployed into the environment using the algorithm is composed of robot units with sensing capability of magnetic anomaly of the earth. A new algorithm is developed for task assignment. It is based on the optimization of weights between robots and tasks. The weights are composed of skill ratings of the robots and pri-orities of the tasks. Multiple teams of mobile units are established in a local area based on these mission vectors. A mission vector is the genetic and gained background information of the mobile units. The genetic background is the inherent structure of their knowledge base in a vector form but it can be dynamically updated with the information gained later on by experience. The mission is performed in a magnetic anomaly environment. The initial values of the mission vectors are loaded by the task assignment algorithm. The mission vectors are updated at the beginning of each sampling period of the motion. Then the teams of robots are created by the support vector machines. A linear optimal hyperplane is calculated by the use of SVM algorithm during training period. Then the robots are clas-siﬁed as teams by use of SVM mechanism embedded in the robots. The support vector machines are implemented in the robots by ordinary op-amps and basic logical gates. Team establishment is tested by simulations and a practical test-bed. Both simulations and the actual operation of the system prove that the system functions satisfactorily.

1. Introduction

In this paper, a new method and algorithm for team establishments within a mobile sensor network (SN) is proposed. The sensor network deployed into the environ-ment using the algorithm is composed of robot units with sensing capability of magnetic anomaly of the earth. They utilize KMZ51 un-isotropic magneto resistive sensors. The sensors are combined to obtain a convolution mask for steepest descent in the magnetic anomaly region. The innovation in this method is to establish more than one

team based on tasks available in the region of operation. A task means that it is a magnetic anomaly of earth’s mag-netic ﬁeld caused by some dangerous mines. The aim is to detect these mines. There may be several magnetic anom-alies within a region. They have to be classiﬁed in an intel-ligent way. This can be achieved in several ways. Why we prefer a new method to identify these sub-regions by mul-tiple robotic teams is that in the previous study[1]it was observed that some of the mobile units had stuck around some sub-regions. For example, inFig. 1, the three yellow1 robots swarm into the sub-region where an AT mine is bur-ied, while the other two blue robots approach into another

⇑Address: Bilkent University, NANO Technology Research Center, 06800 Ankara, Turkey. Tel.: +90 312 290 3050; fax: +90 312 290 1015.

E-mail address:nazlibileksedat@yahoo.com

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For interpretation of color in 1–9 and 14–16, the reader is referred to the web version of this article.

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Measurement

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sub-region where an AP mine is buried. They constituted small groups of robots based on the information gathered by the magneto-resistive sensors mounted on them and calculating the gradient of steepest decent in the magnetic anomalies. There was no external intervention to the mo-tions of the robots. They performed their behavior autono-mously. The behavior of the mobile sensor units are synchronized in time by a time division multiple access (TDMA) method[1]. Some of these groups detected impor-tant objects but the others could not. The robots within the mobile SN had been separated in an inappropriate manner. This caused a waste of resources in most of the cases. It motivated us to create teams of robotic units in an appro-priate way based on the importance of sub-regions. In this paper, we modeled the requirement as a task assignment problem.

In recent years, mobile sensor networks are used in a wide variety of applications such as establishing forma-tions, imitating the behavior of some animals, detecting objects, performing search and rescue activities, area cov-erage, surveillance and reconnaissance, or controlling streetlights for energy savings. In this study we try to achieve autonomously creation of more than one robot teams allocated into several tasks available within an area of operations where more than one anti-tank (AT) and/or anti-personnel (AP) mines exit. The work is motivated by the detection of these buried mines (anti-personnel and anti-tank) at the border regions for clearing purposes. The mines buried across these regions are hard to ﬁnd be-cause of the fact that there are no plans available any more or the mines had changed their locations as a result of some geological, natural and/or manmade effects. In our application, an area of several square meters, normally 1.5 m 1.5 m, is scanned by magneto-resistive sensors and the buried objects are detected by the anomaly of the measured earth magnetic ﬁeld.

Within the scanned region, there may be more than one buried objects creating magnetic anomalies. In our previ-ous study, we classiﬁed the objects one by one based on the data collected by mobile sensors acting over the region

[1]. It was an effective approach for mine detection, but as we observed during operation of the mobile sensor

net-work that some of the robots were grouped around some other buried objects if available within the same region. This is a condition that can frequently been encountered in real applications, therefore we think that we can ap-proach the problem in a different way.

The sensor network deployed into the environment using the algorithm is composed of robot units with sens-ing capability of magnetic anomaly of the earth. They uti-lize KMZ51 un-isotropic magneto resistive sensors. The sensors are arranged in a 3 3 sensor matrix to implement a convolution mask for steepest descent in the magnetic anomaly region. The mask is used to determine the gradi-ent of the ﬁeld as a hardware elemgradi-ent. The robots perform rotation and translation motion at the end of each opera-tion period based on the direcopera-tion of the gradient vector. The mobile units are synchronized by using the method called time division multiple access (TDMA). In this meth-od, a time slot is assigned to each robotic unit to allow them to make movements within its time slot.

We can think of the problem as a task allocation prob-lem which has received signiﬁcant interest in recent years. As seen inFig. 2, the objects found in the region can be con-sidered as tasks and the aim will be to assign multiple ro-bots to these tasks based on an appropriate technique.

There are two common methods applied for task alloca-tion, namely, behavior-based[2–5]and market-based ap-proaches [6,7]. One of the earliest behavior-based method is the so called the ALLIANCE Efﬁciency Problem (AEP) which is an NP-Hard problem[3]. The other famous behavior-based architecture is Broadcast Local Eligibility (BLE) presented in[4]. In general, behavior-based method is a control methodology in which mobile agents are con-trolled through the principled integration of a set of inter-acting behaviors in order to achieve desired system-level behavior. Behavior-based approaches are an extension of reactive architectures and also fall between purely reactive and planar-based extremes[5].

The market-based (or frequently called auction-based) approach is another well known method for solving task allocation problem. The famous examples of marked-based methods are the M + system in[6]and the MURDOCH in

[7]. These methods are based on or a variant of the well-Fig. 1. Groups of robotic units approaching two separate objects.

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known Contract Net Protocol (CNP)[8]. A survey on mar-ket-based approaches can be found in[9], Complex task allocation problems are dealt with in[10]and[11]. In gen-eral, the market-based task allocation approach is based on the principles of a market economy that can be applied to multi-robot coordination where each robot acts to maxi-mize its individual proﬁt and simultaneously improves the efﬁciency of the team. In this approach, two roles are played dynamically by robots: auctineers and bidders

[9,10,12]. The auctioneer is the robot or central station in charge of announcing the tasks and selecting best bid from the bidders. The best bid is the one with lowest cost. The cost is equal to the distance from the robot to the task. The bid is a quantity that reflects how much it will cost the robot to go to a certain waypoint, such as the Euclidean distance or the traversability index. The properties of the market-based task allocation can be listed as follows: As teams of robots as participants in a virtual economy; ro-bots are contracted to complete required tasks in exchange for payment; each robot has well-defined cost and revenue functions to compute the expected gains and losses for performing tasks; they work to maximize their individual profits; in a market, trading tasks via auctions; auctions call for bids; the robots that can perform the tasks for the best price are awarded the resulting contracts.

Another popular method for task allocation is the use of vacancy chains[13,14]. The vacancy chains (VCs) are a re-source distribution strategy common in human and animal societies. Vacancy chains algorithm is uses local task selec-tion, reinforcement learning for estimation of task utility and reward structures based on the vacancy chain frame-work. Three requirements are listed for resource distribu-tion through VCs: the resource must be used by only one individual; a vacancy is required before an individual takes a new resource unit; and vacancy resource units must be scarce and many individuals must occupy sub-optimal unit. In[13], it is demonstrated that VCs could be used to optimize the performance of a group of robots when the

given task conforms to the above three requirements. In fact, VCs often disappear when information is widely avail-able, because information is likely to attract applicants who engage in a competition [14]. A vacancy may be caused by a retirement, which triggers a chain of vacancies through subordinates. At the beginning only one subordi-nate is available. However, when two or more equally qualiﬁed persons apply for the same position, the resource is allocated by the labor market. The fact that a VC is iden-tiﬁed does not imply that it is the prime move of resource allocation. This distinction has never been explicitly stated in empirical research on VCs. All empirical investigations have focused on situations where VCs, at least in their initial steps, did operate as allocation mechanisms. Scarce resource is allocated by means of VCs.

In[2], it is claimed that the multi-robot task allocation could be reduced to an instance of the Optimal Assignment Problem (OAP)[15]that can be casted as a linear optimiza-tion problem.

A well-known method for task assignment called the ‘‘Hungarian Method’’ is given in[16]. It can solve the OAP faster. It develops a computational method that uses the dual linear program in a particularly effective manner.

The simple task assignment problem stated in[16]is as follows: n individuals (denoted by i = 1, 2, . . . , n) are avail-able for n jobs (denoted by j = 1, 2, . . . ,n). They qualify for jobs represented effectively by a (nxn) qualiﬁcation matrix Q in which the rows stand for individuals and columns for jobs and the entries qij= (1 or 0) representing ratings indi-cating that a worker is qualiﬁed or not respectively. Then the simple assignment problem asks: What is the largest number of 1’s that can be chosen from Q with no two cho-sen from the same row or column?

The general assignment problem is as follows: Suppose n individuals (i = 1, 2, . . . ,n) are available for n jobs (j = 1, 2, . . . , n) and that a rating matrix R ¼ ðrijÞ is given, where rijare positive integers, for all i and j. An assignment consists of the choice of one job jifor each individual i such Fig. 2. Basic approach for task assignment to multiple robotic teams.

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that no job is assigned to two different men. The General Assignment Problem asks: For which assignments is the sum of the ratings ðr1j1þ r2j2þ þ rnjnÞ largest?

The Hungarian method and the other task assignment methods mentioned above are generally dealing with the assignment of a single individual to a single job.

We have two kinds of requirements, namely we can either assign single robot to a single task or multiple robots to a single task. In most of the cases, the latter requirement is more frequent. In our application, we want to achieve a collective behavior for detecting any object found in the searched region. We want multiple robots to be swarmed autonomously into an area where a magnetic anomaly cre-ated by a ferromagnetic object is available. Every anomaly can be considered as an area where a mission will be per-formed. Therefore, an anomaly can be considered as a task to be assigned to multiple robots. Since there may be more than one object (i.e., task) in the region, we need to have more than one robot team in the region of operation. Therefore, we need to create a couple of robots teams based on the number of tasks available in that region. We have to classify the teams of robots as separate teams having speciﬁc tasks to be performed. That means that we need to solve a task assignment problem to multiple teams. Each team will have a speciﬁc task. The teams will be established depending on the priorities of the tasks and the skill ratings of the robots. That is, each robot ri, i = 1, . . . , n will have a skill ratings si; i ¼ 1; . . . ; n and each task tj; j ¼ 1; . . . ; m will have a priority, pj; j ¼ 1; . . . ; m.

Every robot in the mobile SN is initially assigned to a task. We developed a new algorithm for initial assignment of the tasks. After initial assignment, the teams are estab-lished by use of support vector machine (SVM) technique. This will speed up the establishment of teams within the mobile SN. Also, it will help the robots to update their tasks periodically during the course of the actions. The duration of the period can be determined a priori. This kind of updating action may improve the overall performance of the system in terms of performance criteria determined in the algorithm. We call this algorithm as the ‘‘mod-med-ian’’ algorithm. Since it sorts out the unworthy robots the related task based on mod and median of the performance values of the robots, and it assigns robots deserving this task.

The structure of this paper is as follows: Section 2 gives the problem deﬁnition in detail and the algorithm develop-ment for task assigndevelop-ment. In Section 3, methods of team establishment are explained. The mission vectors are cre-ated by two methods, namely, the Skill rating – Priority (SP) Method and the Most Skillful Robot (MSR) Method. The theory and implementation of support vector ma-chines are also given in this section. Section 4 gives the experimental results. The conclusion is given in Section 5.

2. Problem deﬁnition and algorithm development 2.1. Problem deﬁnition

The situation is that there are n robots and m tasks within a mission region. The robots have some skills

expressed as skill ratings and the tasks have some priori-ties. As it is stated, the aim is to assign groups of robots to appropriate tasks such that the overall performance be-comes optimum. To do this, an algorithm called the ‘‘mod-median’’ algorithm is developed. The algorithm has two versions. The ﬁrst one makes an assignment in such a way that a single robot is assigned in a single task. No other robots are assigned to the same task. The second version makes an assignment such that multiple robots (deserved ones) can be assigned to a single task in optimum way.

The robot task assignment problem can be stated as fol-lows: Given robots with skill ratings ðri;siÞ; i ¼ 1; . . . ; n and tasks with priorities ðtj;pjÞ; j ¼ 1; . . . ; m, assign robots to tasks so as to maximize the overall expected performance. Normally, the expected performance is the weighed sum of the utilities, which is a concept borrowed from economics, that is the internal estimation of the value or the cost of execution of an action by an individual, and priorities belonging to the tasks.

The course of action by robot teams is divided into time periods, T. One period (T) is also divided into two parts as initial assignment (Ti) and action (Ta), where T = Ti+ Ta. In the initial assignment sub period, the task assignments are achieved by the so called ‘‘mod-median’’ algorithm to be developed here. In the action sub period, (Ta) the robots will be grouped into teams by means of SVMs. At the end of the time period T, a new period with the same structure begins again. The action period is cho-sen long enough such that robots in a team can do several rotational and translational motions within the region of anomaly. The assignments can be updated again at the beginning of the new period. The duration for T can be a time interval passing at least between the start of action and the ﬁrst change in rotation angle of any robot within the team. Normally, it is determined by the user based on previous experiences.

The map of ri’s to tj’s is shown inFig. 3. In this mapping, the weights are the sum of skills of robots si’s and the pri-orities pj’s, that is, wij= si+ pj, i = 1, . . . , n and j = 1, . . . , m. That is, both the skills and the priorities of the tasks must be high. The performance in this problem is the sum of the weights,Pijwij. Assign robots to tasks so as to maximize the overall expected performance (i.e., the sum of the weights), that is, maxPijwij.

If we assign a single robot to a single task, and if n > m, then some robots will be idle, else if n < m, then some of the tasks will be empty. But, in any case, the performance must be greatest in single-to-single assignment. Let’s de-ﬁne the weight matrix as (boldface capital letters represent matrix quantities and boldface small letters represent vec-tor quantities): W ¼ w11 w12 . . . w1m w21 w22 . . . w2m . . . . wn1 wn2 . . . wnm 2 6 6 6 4 3 7 7 7 5 ð1Þ

In case we assign multiple robots (or all of them) to a single task, then we have to determine which mapping will give the greatest performance. That is, among the column vectors:

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w1¼ ½w11w21. . .wn1T;w2¼ ½w12w22. . .wn2T; . . . ;wm

¼ ½w1mw2m. . .wnmT ð2Þ

We can determine which one is the greatest. Then we assign all the robots to that task. However, in this situation, we have assigned all of the robots only to one single task. This is not our aim. We try to assign only the robots that deserve mostly to that job and eliminate the others. How do we achieve such elimination? This can be done by sort-ing the elements of the vectors wj¼ ½w1jw2j. . .wnjT from the greatest to the smallest. Then, we can assign the ones above the median to the tasks and leaving the unassigned ones as idles. What will happen to the idle robots? They will wander around the region and search targets around them. In some time, they may gain a skill, for example approaching a target and deserve a job. Since we do initial task assignments at the beginning of each period T, we re-view the task assignments again and we start a new cycle. In this way, the task assignment becomes a dynamic assignment.

We can show this approach by an example given in the following.

2.2. Example

Let the number of robots be 3 and the number of tasks be 2. The robot-skill and task-priority pairs are (r1, s1) = (1, 5), (r2, s2) = (2, 3) and (r3, s3) = (3, 1), and (t1, p1) = (1, 5) and (t2, p2) = (2, 2). (a) Assign single robot to single task; (b) assign multiple robots to single task. The matrix: W ¼ ½ w1 w2 ¼ w11 w12 w21 w22 w31 w32 2 4 3 5 ¼ ðsðs12þ pþ p11Þ ðsÞ ðs12þ pþ p22ÞÞ ðs3þ p1Þ ðs3þ p2Þ 2 4 3 5 ¼ 10 7 8 5 6 3 2 4 3 5

(a) The single robot to single task assignment: The larg-est weight mapped to t1 is w11= 10, therefore the robot-task pair is (r1, t1). Similarly, the largest

weight mapped to t2is w12= 7. This weight belongs to r1. However, r1is already assigned to t1, therefore it is discarded. The next larger one is w22= 5. There-fore, the assignment pair is (r2, t2). Notice that, ini-tially it is determined which of the applications is the largest.

(b) Multiple robots to single task assignment: The ele-ments of the vector w1 is sorted from largest to smallest as w1¼ ½10 8 6T. The median is w21= 8. Take w11 and w21which are above the med-ian (included). Assign (r1, t1) and (r2, t1). Now, sort the second column vector, w1¼ ½7 5 3T. The median is w22= 5. But, the robot r1 with largest weight w21= 7 has already been assigned to t1and the robot r2 in the mapping w22 has already been assigned to t1. Therefore, we have to delete these assignments. Only remaining assignment is w32. But, w32= 3 is below the median. Therefore, the robot r2cannot be assigned to a task. Only the robots with mapping greater than the median can be assigned to an appropriate task. In this case, r2is idle and t2is empty.

2.3. Algorithm development

In this work, we develop two versions of the assignment algorithm. In the ﬁrst version, a single robot to single task (SRST) assignment can be achieved.

The SRST assignment algorithm is given below: Step 1. Create the (m n) W ¼ ½w1 w2 wm1 wm where wj’s are column vectors whose entries are the weights (that are the sum of skill siand priority pj related to robot-task pair [si, pj] respectively) of map-ping wij= si+ pj, m is the number of tasks. The dimen-sion of the column vectors n is the number of robots. Step 2. Sort the column vectors from the largest to smallest.

Step 3. Get the greatest among the column vectors wj. Step 4. Find the largest element wijof the vector wj. Step 5. Perform the mapping of i to j.

(r , s ) 1 1 (r , s ) 2 2 (r , s )_n _n (t , p ) 1 1 (t , p ) 2 2 (t , p )_m _m w 11 w₁₂ w21 w 22 w 1m w 2m w n1 w_n2 w_nm

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Step 6. Delete wijof the [ri, tj] pair (i.e., discard riand tj). Step 7. If all the assignments are completed, then stop. Step 8. Increment the index of column vector.

Step 9. Go to Step 3.

The second version of the algorithm can perform multi-ple robots to single tasks (MRST) making the overall per-formance maximum. This MRST algorithm is called ‘‘mod-median’’ algorithm. The steps for the mod-median MRST algorithm are given in the following.

MRST Assignment Algorithm:

Step 1. Create the (m n) W ¼ ½w1 w2 wm1 wm where wj’s are column vectors whose entries are the weights (that are the sum of skill siand priority pj related to robot-task pair [si, pj] respectively) of map-ping wij= si+ pj, m is the number of tasks. The dimen-sion of the column vectors n is the number of robots. Step 2. Sort the column vectors from the largest to smallest.

Step 3. Get the greatest among the column vectors wj. Step 4. Find the median element wijof the vector wj. Step 5. Perform the mapping of i to j.

Step 6. Delete riand all of the connections, wij‘s of the ri to the tasks.

Step 7. If all the assignments are completed, then stop. Step 8. Increment the index of column vector.

Step 9. Go to Step 3.

The skill ratings and priorities must be set before the operation or during the operation. They are either set by the user before the operation or determined automatically during the course of action. For autonomous operations, the initial values of the skill ratings and priorities are set to zero.

The skill ratings are related to the distances from the ro-bots to the tasks. The shorter the distance is, the more skill-ful the robot. Hence, the skill ratings can be chosen as the inverse of the distance between the ith robot and the jth task, i.e.,

sij¼

1 dij

: ð3Þ

Here we use double index to show the skill ratings of any robot with a task. A robot may have different skill rat-ings for each individual task within a region.

The priorities of the tasks are the values mainly deter-mined from the number of robots aiming at the target ob-jects. It is the measure of importance of the task. A priority can be deﬁned as the number of robots aiming to that tar-get multiplied by the inverse of the size of the object. That is, pj¼

m

j n 1 cj ð4Þ where cjis the diameter of the jth target object and

mj

is the number of robots aiming at this target.mj

nis the ratio of the number of robots approaching the target j to the total robots n. Since the weights are the sum of skill ratings and priorities, we take the inverse of the diameter in order

to make the units consistent. In this case, both the skill rat-ings and the priorities are in 1/m. The units are consistent now and they can be added. However, there is a problem with this definition. Although it seem that the importance of the target will increase when a number of robots swarm in a confined region, the priority value will go to infinity when the value of the diameter goes to zero. This leads to a misleading operation in the case of, for example, when there is a small object that has a potential of giving rise to a false alarm. In order to prevent such a problem, the diam-eter value is fixed to a constant value. It is taken as the diameter of an AT mine in this application. Then, the prior-ity is only depended on the number of robots desiring to arrive at the target. The definition of the priority is then:

pj¼

m

j

n 1

c ð5Þ

where c is the diameter of an AT mine. This is the so called ‘‘job area circle (JAC)’’. In this application, JAC diameter is chosen as the diameter of AT mine (Fig. 4).

Job area circle (JAC)

Estimated area circle

(a) The time step is k=0 (Initial values of the

skill ratings and priorities are set to zero)

(b) The time step is k=1

Fig. 4. (a) The robots are arbitrarily located within the area of operation; (b) the robots detect a magnetic anomaly and can make rotations based on the direction of gradient of the ﬁeld. After the rotational motion, forward intersections are performed and the points P1 and P2 are

obtained. Then the distances and the job area are estimated. Based on these values, skill ratings and priorities are updated.

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The distances between the robots and the intersection points are determined and set as the skill ratings. They are put into the databases of the robots in the form of vec-tors. Also the coordinates of the intersection points are put into the database vectors. The database vector is called the ‘‘mission vector (MV)’’. The MV is deﬁned clearly in Section 3. The MV is used for the creation of teams by means of SVM as explained later. The ﬂow chart giving the overall operation is given in the Appendix.

3. Team establishment

In this work, depending on the tasks available in a re-gion of operation, it might be necessary to establish more than one team assigned to the tasks. The teams act to accomplish the tasks to which they are assigned. A team is a collection of robots with similar background informa-tion. In this context, the background information is deﬁned in the following sub-section.

3.1. Mission vector of the robots in a mobile SN

The background information of the mobile sensors (MSs) in a mobile SN is the mission vectors (MVs) embed-ded into them. This can be any type of information repre-senting the characteristics of a robot. For example, it may be various attributes assigned to it such as a mission or a task, a friendship code, gender code, or any kind of nature given to it. The background information represented as a vector is deﬁned as

xi¼ ½xi1 xi2 xim1 ximT; i ¼ 1; . . . ; n ð6Þ In this work, we deﬁne the components of a MV in two ways. One is the simple case where the components of a MV are the weights of the background information. In this representation, the skill ratings and priorities of tasks play an important role for constituting the MV. We call this method as ‘‘MV with Skill rating-Priority’’. The repre-sentation of the MV takes into account the average skill ratings of all of the robots within the region of operation. We call this methods as ‘‘MV with the most skillful robot’’.

3.1.1. MV with skill rating – priority (SP) method

This method is based on representing the MVs by the skill ratings of the robots and the priorities of the tasks. A mission vector is the background information of the robot. The background information (weight) is the as-signed missions to the mobile sensor units. As you remember, the mission has two elements, namely the skill ratings of the robots and the priorities of the tasks or targets. That is, the components of the MV are the weights,

xij¼ wij¼ siþ pj; ð7Þ

of the ith robot, for j = 1, ... , m. Where m is the number of tasks in the region of operation. The boundaries for the components of the vector xican be found as follows. Any component of xican be written from Eqs.(3)and (4) as

xij¼ 1 dij þ

m

j cj n ð8Þ Eq. (8) can be written as

xij¼ dij

m

jþ cj dijcj ð9Þ Now, as dij! 1, xij¼

m

j cj n ð10Þ

That means that as the distance increases, the dominant parameter determining the task is the number of robots in the target. As dij?0,

xij¼

1

dij! 1 ð11Þ

This means that as the distance decreases, the dominant term is the skill ratings of the robot.

3.1.2. MV with the Most Skillful Robot (MSR) method In this method, the MV is the difference between the skill rating of the robot and the task minus the mean of the skill ratings between that robot and all the tasks[12]:

xij¼ sij

Xm k¼1

sik

m ð12Þ

where m is the number of tasks. The rationale is that the robots are more likely to win tasks that have a low skill ings for the rest of the team, but a relatively high skill rat-ing for itself.

In this method, assume that the robot rkhas won the tasks tiand tj. The robot rkwill keep tiif and only if

ski Xn l¼1 sli n ! > skj Xn l¼1 slj n ! ð13Þ The meaning of Eq.(13)is that the skill rating of the ro-bot winning the task is greater than the mean of the skill ratings of all the others. That is, the robot chooses the task taking into account the team members. The task chosen actually is the best for the team not just for itself.

In this work, we choose a two dimensional vector for the simplicity to apply to the proposed concept. That is,

xi¼ ½ x11 x12T ð14Þ

The first component represents the primary mission and the second component represents the secondary mis-sion. They are automatically updated during the course of action based on the algorithm given above. If x1> x2, then the mobile sensor performs the first mission, otherwise it performs the second mission. Classification of the mobile sensors as separate teams is based on their missions. If the mobile sensor has to do the first (second) mission, then it must belong to the first (second) class (i.e., team). There-fore, the background information can give an opportunity to classify the mobile sensors as mission teams. Any MS

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can do either a primary mission assigned to it or a second-ary mission assigned to it. That is, classiﬁcation of the mo-bile sensors as mission teams is based on these vectors.

As an example, we can demonstrate the methods by using one of our simulation results depicted inFig. 5. In this simulation, six robots in total are available in the re-gion of operation where one anti-tank (AT) mine, one anti-personnel (AP) mine and one ferromagnetic object are buried. The image is the region of operation obtained by scanning it by the magneto-resistive sensors mounted on the scanner simulating the behavior of the robots. Ini-tially, the MVs are updated on the robots databases embedded inside them. TheTable 1gives the data needed for the analysis.

The data inTable 1is very important. In the application, we discard the third task. So, the MV has two dimensions. Let’s repeat the MVs here again and see the team constitu-tions clearly. For the ﬁrst method where the MV creation is based on the Eq. (8), we have the following MVs (see also theTable 1): x1¼ ½x11 x12T ¼ ½0:066 0:064T x2¼ ½x21 x22 T ¼ ½0:0826 0:07T x3¼ ½x31 x32T ¼ ½0:0526 0:08T x4¼ ½x41 x42T ¼ ½0:0386 0:116T x5¼ ½x51 x52T ¼ ½0:0316 0:15T x6¼ ½x61 x62T ¼ ½0:0286 0:086T

Note that the robots r1and r2go to t1, and the robots r3, r4, r5and r6go to t2. This can be understood from the com-ponents of MVs. When the condition, xi1 > xi2, is satisﬁed, the robot ri will go to t1, otherwise it will go t2.

Similarly, we can look at the results of the Method 2 where the MVs are created based on the Eq. (12). We have the following MVs (see also theTable 1):

x1¼ ½x11 x12T¼ ½0:0253 0:0106T x2¼ ½x21 x22T¼ ½0:033 0:013T x3¼ ½x31 x32 T ¼ ½0:0086 0:0027T x4¼ ½x41 x42T¼ ½0:014 0:03T x5¼ ½x51 x52T¼ ½0:036 0:0484T x6¼ ½x61 x62T¼ ½0:023 0:0007T

Note that in this case the robots r1, r2and r3go to t1, and the robots r4, r5and r6go to t2. This can also be understood from the components of MVs. When the condition, xi1> xi2, is sat-isﬁed, the robot riwill go to t1, otherwise it will go t2. Notice that r3is at the boundary of t1and t2. The components of its MV converge to the values of robots who approach to t2. Although they are so small compared to the values of the robots of t1, it is still in t1. However, it may go either to t1 or t2. We can easily interpret that both of the methods for the constitution of MVs functions very well.

The distribution of mission vectors inside a 2-dimen-sional space is shown inFig. 6. As seen inFig. 6, the MVs are the vectors distributed inside circular regions. Fig. 6

shows the probability distribution function of the MVs

PðxÞ ¼ Pðx11;x12Þ ¼ Ae ðx11x110Þ 2 2r2 x11 þðx12x120Þ 2 2r2 x12 ð15Þ

Fig. 7illustrates the two dimensional space of the MVs and their distributions. In this example, there are two clas-ses of teams created for this application. The densities of

AT mine AP mine Ferromagnetic object Method 1 (for MV) Method 2 (for MV) Dimensions of RoO (135 cm x 70 cm) n=6 (Number of Robots) n=3 (Number of Tasks)

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the vectors as concentric circles in the 2-dimensional space are illustrated inTable 2andFig. 7. As seen, the vectors are concentrated in the circle with r 6 0.3 units at some in-stant of time (Figs. 6 and 7).

3.2. Support Vector Machine (SVM)

In this paper, the robots of mobile SN are aimed to be separated into different teams. This separation is based on the missions assigned to the robots. The missions of ro-bots can constitute a part of their genetic infrastructure. Their genetic infrastructures can be implemented as the mission vectors as described in Section 3. The mobile sen-sors (MSs) with similar background information (that is, mission vectors which are close to each other) come to-gether to form a mission team. Therefore, any MSs must identify the other MSs based on their background informa-tion. For the classification of background information, a simple method called support vector machine (SVM) is used[17]. SVMs are a set of related supervised learning methods used for classification. In simple words, given a set of training examples, each marked as belonging to one of the two categories, an SVM training algorithm builds a model that predicts whether a new example falls into one category or the other[17–21]. This method well suits to our application. A hyperplane, H, can easily be determined for separating the regions where the back-ground genetic information vectors are concentrated. It is noted that this hyperplane is an optimum separation hyperplane. In Fig. 8, there are two classes of vectors, namely, Class 1 and Class2. The vectors, sv1 and sv2are the support vectors. H1and H2are the hyperplanes passing through the support vectors sv1and sv2respectively and parallel to the separation hyperplane H. H1identifies the class 1 and H2identifies class 2.

The vector w determines the hyperplane H. Notice that w is perpendicular to H. The components of the vector w seen inFig. 8will be determined by the Example D given below.

Consider the classiﬁcation of two classes of vectors that are linearly separable (seeFig. 7). The linear classiﬁer is the hyperplane H:

wT_{x þ b ¼ 0} _ð16Þ

with the maximum width (distance between the hyper-planes H1and H2drawn by dotted lines). The first term is the dot product of two vectors w and x. The second term is a scalar variable, b, representing the shift of the hyper-plane from the origin of the reference frame. We have to find the set of pairs (w, b) that characterizes the linear clas-sifier satisfying: 0.1 0.2 0.3 0.4 0.5 0.36 0.27 0.18 0.09 0.05 P(x)

Fig. 6. Probability distribution of classiﬁcation vectors in 2-dimensional space. Table 1

Initial assignment of mission vectors (MVs).

Distance (cm) Skill ratings Mission vectors (MVs) Method 1 xij¼d1ijþ

mj

cjn

Mission vectors (MVs) Method 2 xij¼ sijPmk¼1smik d11 20 d12 70 d13 96 s11 0.05 s12 0.014 s13 0.01 x11 0.066 x12 0.064 x11 0.025 x12 0.01 d21 15 d22 50 d23 78 s21 0.066 s22 0.02 s23 0.013 x21 0.082 x22 0.07 x21 0.033 x22 0.013 d31 28 d32 33 d33 60 s31 0.036 s32 0.03 s33 0.016 x31 0.052 x32 0.08 x31 0.008 x32 0.0027 d41 45 d42 15 d43 43 s41 0.022 s42 0.066 s43 0.02 x41 0.038 x42 0.116 x41 0.01 x42 0.03 d51 65 d52 10 d53 25 s51 0.015 s52 0.1 s53 0.04 x51 0.031 x52 0.15 x51 0.03 x52 0.0484 d61 83 d62 28 d63 17 s61 0.012 s62 0.036 s63 0.058 x61 0.028 x62 0.086 x61 0.02 x62 0.0007

Fig. 7. Distribution of the mission vectors in hyperspace. Legend: H: Hyperplane separating two classes; w: The normal vector of the hyperplane.

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yi¼ wT x þ b ð17Þ yo¼ ciyiP1 ð18Þ where ci2 f1; 1Þg ð19Þ A Lagrangian is given as f ¼1 2kwk 2 þX n i¼1

a

iðciyi 1Þ ð20Þ

where

ai

’s are non-negative Lagrange multipliers. We must now minimize f given in Eq.(21)with respect to w and b simultaneously. This requires that the derivatives of f with respect to all the

ai

’s vanish, all subject to the constraints

ai

P0. Now, this is a convex quadratic programming prob-lem, since the objective function is itself convex, and those parts that satisfy the constraints also form a convex set. The solution can be expressed in terms of linear combina-tion of the training vectors as

w ¼X

n

i¼1

a

icixsvi ð21Þ

It is known that the square of the norm of a vector is

kwk2¼ wT_w _ð22Þ

By putting Eq.(22)into Eq.(21), f ¼3 2 Xn i¼1 Xn j¼1

a

i

a

jcicjxTsvi xsvjþ Xn i¼1 ðaiciÞ b Xn i¼1

a

i ð23Þ Minimize f with respect to

ai

and b subject to

a

iP0 ð24Þ And Xn i¼1

a

ici¼ 0 ð25Þ That is, df dai ¼ 0; i ¼ 1; 2; . . . ; n ð26Þ df db¼ 0 ð27Þ For n = 2, f ¼3 2

a

1

a

1c1c1x T sv1 xsv1þ 3a1

a

2c1c2xTsv1 xsv2 þ3 2

a

2

a

2c2c2x T sv2 xsv2þ

a

1c1b þ

a

2c2b

a

1

a

2 ð28Þ Minimize f: df da1 ¼ 3a1c1c1xTsv1 xsv1þ 3a2c1c2xTsv1 xsv2þ c1b 1 ¼ 0 ð29Þ df da2 ¼ 3a1c1c2xTsv1 xsv2þ 3a2c2c2xTsv1 xsv2þ c2b 1 ¼ 0 ð30Þ df db¼

a

1c1þ

a

2c2¼ 0 ð31Þ Find

a

1¼

a

2 and b for c1¼ 1; c2¼ þ1; xsv1 ¼ ½x11 x21T;xsv2¼ ½x12 x22T:

After ﬁnding the Lagrange multipliers satisfying above rules, an optimum hyperplane H can be plotted perpendic-ular to the weighting vector w ¼Pni¼1

ai

cixsvi(see Eq.(22)). The classiﬁcation can be achieved by the dot product of any vector x to be classiﬁed by the vector w, that is, y = wT

x. If y > +1, x belongs to class 1 else it belongs to class 2. Table 2

Mission vector distribution.

Class 1 (Team 1) Class 2 (Team 2)

Diameter No. of vectors Probability Diameter No. of vectors Probability

60.10 20 0.36 60.10 21 0.38 60.22 15 0.27 60.22 16 0.29 6_0.30 ₁₀ _0.18 6_0.30 ₉ _0.16 60.70 5 0.09 60.70 4 0.07 60.95 3 0.0054 60.95 2 0.037 >1.0 2 0.0036 >1.0 2 0.037

w

1 1.5 2 2.5 3 3.5 x1 x2 H H1 H2 sv1 sv2 0.413 -0.367 -1 +1 Class 1 Class 2 1 2 3 4 4

Fig. 8. The distribution of vectors inside a 2-dimensional space. Legend: svi: Support vectors (i = 1, 2); H: Hyperplane; w: Normal vector to the

hyperplane representing it; xi: ith coordinate of the space; Hi: Hyperplane

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3.3. Example

Let’s the support vectors be

xsv1¼ ½1:5 2:3T; xsv2¼ ½2:4 1:5T; c1¼ 1 and c2

¼ 1

Find the hyperplane that can separate the two dimensional space into two classiﬁcation regions. The Eq.(24)can be rewritten as f ¼3 2 Xn i¼1 Xn j¼1

a

i

a

jcicjxTsvi xsvjþ Xn i¼1

a

ici ! b X n i¼1

a

i For n = 2, f ¼1 2½a1c1x1 T_þ

_a

2c2x2T ½a1c1x1þ

a

2c2x2 þ

a

1ðc1½a1c1x1T þ

a

2c2x2T x1þ b 1Þ þ

a

2ðc2½a1c1x1Tþ

a

2c2x2T x2 þ b 1Þ

It can be reduced to Eq.(28): f ¼3 2

a

1

a

1c1c1x T sv1 xsv1þ 3a1

a

2c1c2xTsv1 xsv2 þ3 2

a

2

a

2c2c2x T sv2 xsv2þ

a

1c1b þ

a

2c2b

a

1

a

2 From Eqs.(29)–(31)and putting the support vectors,

22:62a1 21:15a2 b ¼ 1 ð32Þ

21:15a1þ 24:03a2þ b ¼ 1 ð33Þ

df

db¼

a1

c1þ

a2

c2¼ 0 gives

a1

+

a2

= 0, that is,

a1

=

a2

. Putt-ing this result into Eqs.(33) and (34)gives

a

1¼

a

2¼ 0:45977 ð34Þ

and

b ¼ 0:3241 ð35Þ

Then the w vector can be written as (seeFig. 8)

w ¼X

2

i¼1

a

icixsvi¼ ½0:413793 0:367816T ð36Þ

3.4. Implementation of SVM for team establishment The SVM classifier can easily be implemented with operational amplifiers and logic gates as shown inFig. 9. In this application, the mission vectors are 2-dimensional as explained above. This means that there are two classes of MSs in the operation area. It receives the class vectors as inputs. The input stage, which is a summing amplifier, implements the dot product of the input vector with the vector w. It then produces the output signal as yi= wTxi. The next stage is a limiter that decides whether y0is 1 or +1 representing the appropriate class to which it

1 1 -1 -1 +

TTL

Level

1

Fig. 9. SVM classiﬁer for team establishment. [-10.57 15.69] sv2 sv1 H w = 0,12 0,1 0,08 0,06 0,04 0 0 0,01 0,02 0,03 0,04 0,05 0,06 0,07 0,08 0,1055 0,0976 0,0685 _0,0637 x1 x2

Mission vector (Method 1)

T

Fig. 10. The hyperplane H separating two teams of robots. Here, the mission vectors are x1¼ ½0:06926 0:0637T, x2¼ ½0:0566 0:0685T;

x3¼ ½0:037 0:0976T, and x4¼ ½0:0317 0:1055T. The support vectors

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belongs. Since a TTL circuit is used to activate the corre-sponding gate output, it is ﬁrst converted to the TTL level and then it is passed through the logic circuit realized by

the NAND gates. The outputs H1and H2are active low. That is, if Hi= 0, then the robot is a member of ith team (where i = 1, 2).

Table 3

Initial assignment of mission vectors (MVs).

Distance (cm) Skill ratings Mission vectors (MVs) Method 1 xij¼d1ijþ

mj

cjn

Mission vectors (MVs) Method 2 xij¼ sijPmk¼1smik d11 19 d12 73 d13 100 s11 0.052 s12 0.014 s13 0.01 x11 0.069 x12 0.064 x11 0.019 x12 0.019 d21 25 d22 54 d23 78 s21 0.040 s22 0.018 s23 0.013 x21 0.056 x22 0.07 x21 0.010 x22 0.010 d31 49 d32 21 d33 40 s31 0.020 s32 0.048 s33 0.025 x31 0.037 x32 0.098 x31 0.013 x32 0.013 d41 66 d42 18 d43 25 s41 0.015 s42 0.055 s43 0.040 x41 0.031 x42 0.105 x41 0.02 x42 0.02 H w= [13.79 -13.79]T

Mission vector (Method 2)

0,035 0,02 0,015 0,01 0,005 -0,005 -0,01 -0,015 - 0,02 x1 x2 _{-0,025 -0,02} _-0,015 _{-0,01 -0,005} _0,005 _0,01 _0,015 _0,02 0,0202 0,0136 -0,01075 -0,01945

Fig. 11. The hyperplane H separating two teams of robots. Here, the mission vectors are x1¼ ½0:01945 0:01945T, x2¼ ½0:01075 0:01075T;x3¼

½0:0136 0:0136T

and x4¼ ½0:0202 0:0202T. The support vectors are chosen as x3¼ sv1¼ ½0:0136 0:0136T x2¼ sv2¼ ½0:01075 0:01075T.

w = [-6.01 15.45]T H 0,16 0,14 0,12 0,08 0,06 0,02 0 0 0,01 0,02 0,03 0,04 0,05 0,06 0,07 0,08 0,09 0,15 0,116 0,086 0,08 0,064 0,07

Mission Vectors (Method 1)

x1

x2

0,10

Fig. 12. The hyperplane H separating two teams of robots. Here, the mission vectors are x1¼ ½0:0660 0:0640T;x2¼ ½0:0826 0:0700T, x3¼

½0:0526 0:0800T;x4¼ ½0:0386 0:1160 T ;x5¼ ½0:0316 0:1500 T and x6¼ ½0:0286 0:0860 T

. The support vectors are chosen as x3¼ sv1¼

½0:0526 0:0800T

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4. Experimental results

The operation of the system is tested by simulations implemented in Visual C++ environment and also an experimental prototype systems realized as experimental Sumo robots. The magnetic anomalies are created by one real anti-tank (AT) mine, one training anti-personnel (AP)

mine and a ferromagnetic object (a bolt). For illustrative purposes, some of the simulations and practical applica-tion results are given in Figs. 14–17. The locations of anomalies created by the AT mine, AP mine and the ferro-magnetic bolt are considered as the ﬁrst task, the second task and the third task respectively. The task assignments and team establishments processes for each case are

sum-0,05 0,04 0,03 0,02 0,01 0 -0,01 -0,02 x1 -0,03 0 -0,02 -0,03 -0,04 -0,02 -0,01 0,01 -0,04 0,0484 0,03 0,0007 0,0027 -0,0106 -0,013 H w = [21,00 -1,33]T

Mission Vectors (Method 2)

x2

Fig. 13. The hyperplane H separating two teams of robots. Here, the mission vectors are x1¼ ½0:0253 0:0106T;x2¼ ½0:0330 0:0130T, x3¼

½0:0086 0:0027T;x4¼ ½0:0140 0:0300 T ;x5¼ ½0:0366 0:0484 T and x6¼ ½0:0230 0:0007 T

. The support vectors are chosen as x3¼ sv1¼

½0:0086 0:0027T

and x6¼ sv2¼ ½0:023 0:0007 T

.

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marized inTables 1–3. In the experiments, the anomalies of the AT and AP mines are taken as the tasks and the fer-romagnetic object is discarded in order to obtain a two-dimensional mission vectors.

The experiments are carried out by using the following parameters: the diameters of the ﬁrst job assignment cir-cles are c1= 30 cm, c2= 10 cm which are the diameters of AT and AP mines respectively. The two methods, method 1 (MV with Skill rating – Priority (SP) Method) and method 2 (MV with the Most Skillful Robot (MSR) Method) are used for the determination of the mission vectors.

For the ﬁrst experiment (Fig. 10), the number of robots n = 4, the number of tasks m = 2, the number of robots at the vicinity of task 1 is #1= 2, and the number of robots at the vicinity of task 2 is #2= 2.Table 3summarizes the results of the ﬁrst experiment.

FromTable 3, the mission vectors obtained using meth-od 1 (MV with Skill rating – Priority (SP) Methmeth-od) are as follows: x1¼ ½0:06926 0:0637T, x2¼ ½0:0566 0:0685T, x3¼ ½0:037 0:0976T, and x4¼ ½0:00317 0:1055T. The support vectors are chosen as x2¼ sv1¼ ½0:0566 0:0685T and x3¼ sv2¼ ½0:037 0:0976T.

Using Eqs.(29)–(31), the non-negative Lagrange multi-pliers can be calculated as

a1

=

a2

= 539.23 and the offset value b = 2.435. Using Eq.(21), the vector w can be calcu-lated as w ¼ ½10:57 0:1055T. This vector deﬁnes the

hyperplane H separating two classes of mission vectors that is used for the establishment of the teams (Fig. 10).

The mission vectors calculated by the method 2 (MV with the Most Skillful Robot (MSR) Method) are as follows: x1¼ ½0:01945 0:01945T, x2¼ ½0:01075 0:01075T, x3¼ ½0:0136 0:0136T, and x4¼ ½0:0202 0:0202T. For these mission vectors, the hyperplane is shown in

Fig. 11for which w ¼ ½13:79 13:79T and b = 0.116. For the second illustrative experiment (Fig. 14), n = 6, m = 2, #1= 3, #2= 3. The results of the task assignment pro-cess are given inTable 1.

FromTable 1, the mission vectors obtained using meth-od 1 (MV with Skill rating – Priority (SP) Methmeth-od) are as follows: x1¼ ½0:0660 0:0640T, x2¼ ½0:0826 0:0700T; x3¼ ½0:0526 0:0800T, x4¼ ½0:0386 0:1160T; x5¼ ½0:0316 0:1500T _{and x}

6¼ ½0:0286 0:860T. The support vectors are chosen as x3¼ sv1¼ ½0:0526 0:0800T and x4¼ sv2¼ ½0:0386 0:1160T.

Using Eqs.(29)–(31), the non-negative Lagrange multi-pliers can be calculated as

a1

=

a2

= 429.38 and the offset value b = 3.72. Using Eq.(21), the vector w can be calcu-lated as w ¼ ½6:01 15:45T_{. This vector deﬁnes the} hyperplane H separating two classes of mission vectors that is used for the establishment of the teams (Fig. 12).

The mission vectors calculated by the method 2 (MV with the Most Skillful Robot (MSR) Method) are as follows: Fig. 15. Simulation with six mobile sensors grouped automatically into two teams in a magnetic ﬁeld environment.

Fig. 16. Experiments with prototype experimental robots. In these pictures, the two robots detect the anomalies created under the wooden platforms and the motions are monitored on the computer screen.

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x1¼ ½0:0253 0:0106T; x2¼ ½0:0330 0:0130T, x3¼ ½0:0086 0:0027T_, _x

4¼ ½0:0140 0:0300T; x5¼ ½0:0366 0:0484Tand x6¼ ½0:0230 0:0007T. The sup-port vectors are chosen as x3¼ sv1¼ ½0:0086 0:0027T and x6¼ sv2¼ ½0:023 0:0007T.

Using Eqs. (29)–(31), the non-negative Lagrange multipliers can be calculated as

a1

=

a2

= 664.96 and the offset value b = 0.447. Using Eq.(21), the vector w can be calculated as w ¼ ½21:00 1:33T. This vector de-ﬁnes the hyperplane H separating two classes of mission

Locate robots arbitrarily

Set skill ratings and priorities initially to zero (si=0, Pj=0) Move robots one step And make forward intersections Set Job Area Circle

(JAC) Radius Start

Give the same task name

to all points falling inside

the JAC, e.g. tj

Find the distance dij Between robot ri and task tj

Take the distances as Skill ratings

si=1/dij

Determine the priority Value pj=cj x ν cj is diameter of JAC

and ν is the number of points

within the circle

Create W Matrix (wij=si+pj) SRST ? YES SRST Algorithm NO MRST Algorithm End of period T ? YES NO SVM

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vectors that is used for the establishment of the teams (Fig. 13).

And for the last experiment performed by prototype ro-bots, n = 2, m = 2, #1= 1, #2= 1. In this experiment, we tried to see the functioning and performance of the practical system in performing task assignment and team establish-ment by SVM, though the number of robots are very restricted.

5. Conclusion

In this paper, the team establishment problem is solved by using task assignment approach. A new algo-rithm is developed for task assignment. The task assign-ment algorithm has two versions. The first version is used for the assignment of single robot to a single task. However, the second version is used for the assignment of multiple robots to a single task. The algorithm depends on the optimization of weights composed of skill ratings of the robots and priorities of the tasks. The skill ratings are related to the distances from the robots to the tasks. The shorter the distance is, the more skillful the robot. The priorities of the tasks are the values mainly deter-mined from the number of robots aiming at the target ob-jects. It is the measure of importance of the task. A priority can be defined as the number of robots aiming to that target multiplied by the inverse of the size of the object. Two methods are used to define the weights. The weights are used for the creation of mission vectors which constitute the background information of the ro-bots. Depending on the mission vectors, robots are classi-fied as teams.

In this work, the operation of robot teams is performed in a periodic fashion. The period is divided into two stages. The ﬁrst stage is the initialization stage where the initial task assignment is done by the task assignment algorithm developed here. In the second stage, the team establish-ment is achieved by using SVM method which creates the teams based on the mission vectors of the robots ob-tained in the ﬁrst stage of the period. Dividing the course of action in this way facilitates the operation of the teams and also gives an opportunity to update the mission vec-tors during the operation. It reduces the communications needs as well. During the second stage, robots move with-out any communications. They come together as teams by use of the SVM mechanism.

The sensor network created here is a mobile sensor network that is composed of mobile units making rota-tional and translarota-tional motion within a period of opera-tion. Each mobile unit has a capability of sensing the direction of the gradient vector of the magnetic ﬁeld by means of a convolution mask created by a sensor mecha-nism with 3 3 sensor grid. The overall operation is syn-chronized in time by use of a time division multiple access (TDMA) method.

The methods are demonstrated by simulations and a practical example. In order to simplify the proof of the concept, a two dimensional mission vectors are used. The task assignment algorithm developed here has no restriction on the dimension of vectors. It may be applied for assignment of many robotic teams to many tasks. The SVM method is also suitable for separation of multiple regions whether linear of nonlinear fashion. In this application, the SVM approach is adapted for the classiﬁ-cation of two dimensional vectors. In that case, a linear separation hyperplane can easily be found. But it can be extended to multiple and nonlinear separations as well.

The approach developed here gives satisfactory results. It suits very well the solution of the problem for detecting anti-tank and anti-personnel mines buried long years be-fore at the border regions. The method helps the works in-tended to clear this kind of regions.

Appendix A A.1. Illustrations

In this appendix, the outputs of team establishment process are given. The process automatically determines the skill ratings based on the distances to the tasks and pri-orities of the tasks. These values are used to create the mis-sion vectors of the robots. They move in the potential ﬁeld and constitute the teams based on these vectors by means of SVMs. The mission vectors are periodically updated dur-ing the course of action. The ﬁrst illustration shows the behavior of four robots in a magnetic anomaly environ-ment. In the second illustration, the number of robots is in-creased to six. The third illustration is the conceptual demonstration of the proposed approach by two practical experimental robotic systems. The results are encouraging (seeFigs. 14–17).

A.2. Flowchart

This appendix gives the ﬂow chart of the overall opera-tion. It includes the team assignment algorithms as well. References

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